Here are some quick tips to increase your Alexa rank.
Install the Alexa toolbar and set your blog as your homepage. This is the first step that you just HAVE to do.
Make others to use the Alexa toolbar- Friends or readers, just make them have the toolbar.
Write content that is related to webmasters or technology- There is a very high chance that people in these niches will have the Alexa toolbar installed.
Feel free to share more tips.
PS. I know there are people who say that Alexa has changed its algorithm, but user Sensonize on the forum and I have confirmed it works, with a case study.